An equivariant index formula for elliptic actions on contact manifolds

TL;DR

This paper develops an index formula for elliptic group actions on contact manifolds, linking geometric structures with equivariant indices using advanced differential forms and Chern character techniques.

Contribution

It introduces a natural equivariant differential form associated with contact structures and derives an explicit index formula involving this form and the Chern character.

Findings

Derived an explicit index formula for elliptic actions on contact manifolds.

Connected contact geometry with equivariant index theory using generalized differential forms.

Utilized the Chern character with compact support to facilitate the index computation.

Abstract

Given an elliptic action of a compact Lie group $G$ on a co-oriented contact manifold $(M,E)$ one obtains two naturally associated objects: A $G$-transversally elliptic operator $\dirac$, and an equivariant differential form with generalised coefficients $\mathcal{J}(E,X)$ defined in terms of a choice of contact form on $M$. We explain how the form $\mathcal{J}(E,X)$ is natural with respect to the contact structure, and give a formula for the equivariant index of $\dirac$ involving $\mathcal{J}(E,X)$. A key tool is the Chern character with compact support developed by Paradan-Vergne \cite{PV1,PV}.